# Dmitri Kozlov

Architect, scientist

Research Institute of the Theory and History of Architecture and Town-planning of Russian Academy of Architecture and Building Sciences

Moscow, Russia

Knots have been the subject of traditional art since ancient time. Woven patterns, stone and wood carving and other 2D knotted ornaments were very popular throughout the world. Today artists depict knots mainly as 3D objects to express the interaction of knots with the surrounding space.

My approach is based on the idea that cyclic periodic knots made of resilient filaments like steel wire behave as kinetic structures. Knots tied with such materials must have a large number of physically contacting crossings namely the vertices of surfaces. The crossings slide along the resilient filaments which at the same time twist around their central axis. The waves on the filaments move and change their lengths to adapt to the current disposition of the contact crossings. Thanks to these properties the knots change their geometry as a whole and create vertex or point surfaces with an arbitrary Gaussian curvature. I designated as NODUS-structures the complicated knots of this type.

My approach is based on the idea that cyclic periodic knots made of resilient filaments like steel wire behave as kinetic structures. Knots tied with such materials must have a large number of physically contacting crossings namely the vertices of surfaces. The crossings slide along the resilient filaments which at the same time twist around their central axis. The waves on the filaments move and change their lengths to adapt to the current disposition of the contact crossings. Thanks to these properties the knots change their geometry as a whole and create vertex or point surfaces with an arbitrary Gaussian curvature. I designated as NODUS-structures the complicated knots of this type.

Spherical NODUS-structure “Homage to Gauss”

50x50x50 cm

steel spring wire

2007

This transformable spherical NODUS structure is based on the principle of a complicated periodic knot known as chain Turk’s-Head. It has 17 loops and two 17-gons at the polar openings. I chose the number 17 because it symbolizes the early discovery of Karl Friedrich Gauss that states 17-gon can be constructed with a ruler and compasses.

The name of Gauss is closely connected with the beginning of knot theory and the theory of surfaces. As opposed to solid surfaces that can not change their curvatures without breaks and folds, the point surfaces of NODUS-structures can be changed from the positive Gaussian curvature (elliptic) to the negative one (hyperbolic) through the neutral (parabolic) curvature and vice versa due to the torus rotation. NODUS-structures, as outer (elliptic) or an inner (hyperbolic) parts of torus, can rotate around the imaginary circle axis. The structure can turn inside out and take the forms of the outer and the inner parts of the torus surface.

The name of Gauss is closely connected with the beginning of knot theory and the theory of surfaces. As opposed to solid surfaces that can not change their curvatures without breaks and folds, the point surfaces of NODUS-structures can be changed from the positive Gaussian curvature (elliptic) to the negative one (hyperbolic) through the neutral (parabolic) curvature and vice versa due to the torus rotation. NODUS-structures, as outer (elliptic) or an inner (hyperbolic) parts of torus, can rotate around the imaginary circle axis. The structure can turn inside out and take the forms of the outer and the inner parts of the torus surface.

Toroidal NODUS-structure “Mirror Torus Knots”

50x50x30 cm

steel spring wire

2008

Any physical knotted filament has a thickness, so any real knot is a torus. A non-knotted torus or simple ring is known as a trivial knot. There is a special class of torus knots that can be placed on the surface of a torus without any self-crossings. The simplest trefoil is an example of torus knots.

A trefoil knot may have two mirror types: a left and a right. Each of them can be tied on the torus surface without self crossings, but when tied together on the same torus, they cross each other and form an elementary knotted fabric on the torus surface. If both knots are made of resilient filaments and their crossings are really contacting, they form the structure of elementary toroidal self-supporting point surface.

My work describes the principle of materialization of two mirror torus knots. I took two mirror torus knots with parameters q=17 and p=16 and wove them into a self-supporting kinetic toroidal NODUS-structure. The structure may take planar and three-dimensional forms.

A trefoil knot may have two mirror types: a left and a right. Each of them can be tied on the torus surface without self crossings, but when tied together on the same torus, they cross each other and form an elementary knotted fabric on the torus surface. If both knots are made of resilient filaments and their crossings are really contacting, they form the structure of elementary toroidal self-supporting point surface.

My work describes the principle of materialization of two mirror torus knots. I took two mirror torus knots with parameters q=17 and p=16 and wove them into a self-supporting kinetic toroidal NODUS-structure. The structure may take planar and three-dimensional forms.

NODUS-structure with one-side surface “Self Crossing Möebius Band”

50x40x30

steel spring wire

2009

Knots are closely connected with one-side surfaces. The edge of a Möebius band made of a ribbon twisted through π angle is a continuous curve namely a trivial knot. Then the same ribbon is twisted through 3π angle its edge is the trefoil knot.

Another approach to the shape of Möebius band arise from the cutting of Klein bottle on two equal parts. These parts have the form of Möebius band with self-crossings.

In turn the Klein bottle can be done as a torus with figure eight cross section twisted through π angle and thereby can be presented by two mirror torus knots as explained in my previous description.

These two mirror torus knots on the imaginary Klein bottle surface can be cut along the line of the equator and divided into two cyclic knots or links with self-crossings.

The same idea is reflected in this my work. A cyclic knot made of a single piece of steel wire forms a NODUS-structure with self-crossing and one edge. It represents a Möebius band as a half Klein bottle.

Another approach to the shape of Möebius band arise from the cutting of Klein bottle on two equal parts. These parts have the form of Möebius band with self-crossings.

In turn the Klein bottle can be done as a torus with figure eight cross section twisted through π angle and thereby can be presented by two mirror torus knots as explained in my previous description.

These two mirror torus knots on the imaginary Klein bottle surface can be cut along the line of the equator and divided into two cyclic knots or links with self-crossings.

The same idea is reflected in this my work. A cyclic knot made of a single piece of steel wire forms a NODUS-structure with self-crossing and one edge. It represents a Möebius band as a half Klein bottle.